Indices: Simplification 3b

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Created this as a copy of Indices: Simplification 3a.

Name	Status	Author	Last Modified
Surds: Simplifying Expressions 1a	Ready to use	Ben McGovern	15/08/2023 15:04
Surds: Simplifying Expressions 1b	Ready to use	Ben McGovern	15/08/2023 15:04
Surds: Simplifying Expressions 2b	Ready to use	Ben McGovern	15/08/2023 15:49
Surds: Simplifying Expressions 2a	Ready to use	Ben McGovern	15/08/2023 15:49
Surds: Simplifying Expressions 3a	Ready to use	Ben McGovern	15/08/2023 15:31
Surds: Simplifying Expressions 3b	Ready to use	Ben McGovern	15/08/2023 15:14
Surds: Simplifying Expressions 4a	Ready to use	Ben McGovern	15/08/2023 15:29
Surds: Simplifying Expressions 4b	Ready to use	Ben McGovern	15/08/2023 15:26
Surds: Simplifying Expressions 4c	Ready to use	Ben McGovern	15/08/2023 15:50
Surds: Simplifying Expressions 5a - Rationalising the Denominator	Ready to use	Ben McGovern	15/08/2023 15:55
Surds: Simplifying Expressions 5b - Rationalising the Denominator	Ready to use	Ben McGovern	15/08/2023 16:07
Surds: Simplifying Expressions 6 - Rationalising the Denominator	Ready to use	Ben McGovern	21/09/2022 16:31
Surds: Simplifying Expressions 7 - Rationalising the Denominator	Ready to use	Ben McGovern	21/09/2022 16:31
Quadratics: Using the Quadratic Formula	Ready to use	Ben McGovern	14/09/2022 16:14
Quadratics: Factorisation 1	Ready to use	Ben McGovern	14/08/2023 15:01
Quadratics: Factorisation 2	Ready to use	Ben McGovern	14/08/2023 15:14
Quadratics: Completing the Square 1	Ready to use	Ben McGovern	14/08/2023 14:59
Quadratics: Completing the Square 2	Ready to use	Ben McGovern	14/09/2022 16:03
Quadratics: Interpreting the Discriminant	Ready to use	Ben McGovern	14/09/2022 16:08
Quadratics: Factorisation 3 - Difference of 2 squares	Ready to use	Ben McGovern	14/08/2023 15:33
Indices: Simplification 1a	Ready to use	Ben McGovern	26/06/2023 12:12
Indices: Simplification 1b	Ready to use	Ben McGovern	26/06/2023 14:31
Indices: Simplification 1c	Ready to use	Ben McGovern	26/06/2023 14:30
Indices: Simplification 2a	Ready to use	Ben McGovern	26/06/2023 14:33
Indices: Simplification 2b	Ready to use	Ben McGovern	26/06/2023 14:36
Indices: Simplification 3a	Ready to use	Ben McGovern	26/06/2023 14:37
Indices: Simplification 3b	Ready to use	Ben McGovern	26/06/2023 14:40
Indices: Change of Notation 1	Ready to use	Ben McGovern	15/08/2023 11:43
Indices: Change of Notation 2	Ready to use	Ben McGovern	15/08/2023 11:51
Indices: Change of Notation 3	Ready to use	Ben McGovern	15/08/2023 11:54
Solve Quadratics - Factorisation 1	Ready to use	Ruth Hand	14/08/2023 15:44
Quadratics: Solving a Quadratic Equation	Ready to use	Ben McGovern	09/12/2022 16:45
Quadratics: Solving by Factorisation	Ready to use	Ben McGovern	14/09/2022 16:25
AD2 Quadratics - solve	Ready to use	Megan Oliver	09/04/2024 15:13
AD5 Difference of two squares	Ready to use	Megan Oliver	09/04/2024 15:14
AA2 Indices - divide	Ready to use	Andrew Neate	09/04/2024 15:00
AA4 Indices - Fractional 2	Ready to use	Andrew Neate	09/04/2024 15:01
AA3 Indices - Fractional 1	Ready to use	Andrew Neate	09/04/2024 15:01
AA1 Indices - times	Ready to use	Andrew Neate	28/03/2024 13:17
Calculation with indices - times	Ready to use	Will Morgan	26/06/2024 11:53
Calculation with Indices - divide	Ready to use	Will Morgan	26/06/2024 12:07

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Question statement

Calculate the following expression without a calculator:

$\frac{\var{a}^\var{n} \times \var{b}^\var{m}}{\sqrt{\var{a^4} \times \var{b^2}}}$

Give any introductory information the student needs.

			Name	Type	Generated Value

a	integer	2
b	integer	3
n	integer	4
m	integer	3

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Advice

To simplify $\frac{\var{a}^\var{n} \times \var{b}^\var{m}}{\sqrt{\var{a^4} \times \var{b^2}}}$ , we want to make use of the following 2 rules:

$\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}$ ;
$\frac{a^n}{a^m} = a^{n-m}$ .

Applying rule 1 to the denominator:

$\begin{split}\frac{\var{a}^\var{n} \times \var{b}^\var{m}}{\sqrt{\var{a^4} \times \var{b^2}}} &\,= \frac{\var{a}^\var{n} \times \var{b}^\var{m}}{\sqrt{\var{a^4}} \times \sqrt{\var{b^2}}} \\\\ &\,= \frac{\var{a}^\var{n} \times \var{b}^\var{m}}{\var{a^2} \times \var{b}}. \end{split}$

Since $\var{a^2} = \var{a}^2$ , we can rewrite the denominator so that the numerator and denominator are both in terms of $\var{a}$ and $\var{b}$ , allowing us to apply rule 2:

$\begin{split} \frac{\var{a}^\var{n} \times \var{b}^\var{m}}{\var{a^2} \times \var{b}} &\,= \frac{\var{a}^\var{n} \times \var{b}^\var{m}}{\var{a}^2 \times \var{b}} \\\\ &\,= \var{a}^{\var{n}-2} \times \var{b}^{\var{m}-1} \\ &\,= \var{a}^\var{n-2} \times \var{b}^\var{m-1} . \end{split}$

Therefore,

$\begin{split} \frac{\var{a}^\var{n} \times \var{b}^\var{m}}{\sqrt{\var{a^4} \times \var{b^2}}} &\,= \var{a}^\var{n-2} \times \var{b}^\var{m-1} \\ &\,= \var{a^(n-2)*b^(m-1)}. \end{split}$

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This question is used in the following exams:

Algebra DIAGNOSYS trial by Ruth Hand in Ruth's workspace.
Indices by Ben McGovern in MASH Bath: Moodle quizzes and TS.
All the good stuff by Chris Templet in Chris's workspace.
New Exam by Ben Tito in Ben's workspace.
Testing of the website by Manpreet Singh in Manpreet's workspace.
Indices 2 by Luke Middleton in Luke's workspace.
MATH2411 - Indices by Blathnaid Sheridan in Blathnaid's workspace.
MATH2411 - Assessment 1 by Blathnaid Sheridan in Blathnaid's workspace.
MATH2411 - CA1 by Blathnaid Sheridan in Blathnaid's workspace.

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Indices: Simplification 3b

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

Feedback

History

Comment

Checkpoint description

Ben McGovern 2 years, 6 months ago

Ben McGovern 3 years, 8 months ago

Ben McGovern 4 years, 8 months ago

Error in variable testing condition

Error

Warning

Generated value: integer

Parts

Mathematical expression

Advanced settings

Checking accuracy

Variable value generators

Pattern restriction

Variables

String restrictions

Generated value: `integer`